Theorem tpssd 32474

Description: Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpssd.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
tpssd.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
tpssd.3	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Assertion

Ref	Expression
tpssd	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpssd

Step	Hyp	Ref	Expression
1		tpssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		tpssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		tpssd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		tpssi 4810	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2109   ⊆ wss 3922  {ctp 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-un 3927  df-ss 3939  df-sn 4598  df-pr 4600  df-tp 4602

This theorem is referenced by:  constrlccllem  33751  constrcccllem  33752  cos9thpiminplylem2  33781